Gradient Estimates for p-Laplacian Lichnerowicz Equation on Noncompact Metric Measure Space

2020 ◽  
Vol 41 (3) ◽  
pp. 397-406
Author(s):  
Liang Zhao ◽  
Ming Shen
Keyword(s):  
Measure Space ◽  
Metric Measure Space ◽  
Gradient Estimates ◽  
Lichnerowicz Equation

scholarly journals Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space

2017 ◽  
Vol 2017-3 (103) ◽  
pp. 19-28
Author(s):  
Luigi Ambrosio ◽  
Nicola Gigli ◽  
Giuseppe Savaré
Keyword(s):  
Ricci Curvature ◽  
Measure Space ◽  
Optimal Transport ◽  
Metric Measure Space

scholarly journals Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Toni Heikkinen
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Space Setting ◽  
Poincare Inequalities

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

A note on fractional integral operators defined by weights and non-doubling measures

2010 ◽  
Vol 106 (2) ◽  
pp. 283 ◽  
Author(s):  
Oscar Blasco ◽  
Vicente Casanova ◽  
Joaquín Motos
Keyword(s):  
Measure Space ◽  
Fractional Integral ◽  
Integral Operators ◽  
Metric Measure Space ◽  
Integral Type ◽  
Lebesgue Spaces ◽  
Lipschitz Spaces ◽  
Fractional Integral Operators ◽  
Doubling Measures

Given a metric measure space $(X,d,\mu)$, a weight $w$ defined on $(0,\infty)$ and a kernel $k_w(x,y)$ satisfying the standard fractional integral type estimates, we study the boundedness of the operators $K_w f(x)=\int_X k_w(x,y)f(y)\,d\mu(y)$ and $\tilde K_w f(x)=\int_X (k_w(x,y)-k_w(x_0,y))f(y)\,d\mu(y)$ on Lebesgue spaces $L^p(\mu)$ and generalized Lipschitz spaces $\mathrm{Lip}_\phi$, respectively, for certain range of the parameters depending on the $n$-dimension of $\mu$ and some indices associated to the weight $w$.

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